On special identities for dialgebras

Abstract

For every variety of algebras over a field, there is a natural definition of a corresponding variety of dialgebras (Loday-type algebras). In particular, Lie dialgebras are equivalent to Leibniz algebras. We use an approach based on the notion of an operad to study the problem of finding special identities for dialgebras. It is proved that all polylinear special identities for dialgebras can be obtained from special identities for corresponding algebras by means of a simple procedure. A particular case of this result confirms the conjecture by Bremner, Felipe, and Sánchez-Ortega [M.R. Bremner, R. Felipe, and J. Sánchez-Ortega, Jordan triple disystems, Comput. Math. Appl. 63 (2012), pp. 1039–1055].